POLARON EFFECT ON THE BINDING-ENERGY OF A HYDROGENIC IMPURITY IN A QUANTUM-WELL

PHYSICAL REVIEWB VOLUME 48, NUMBER 16 15OCTOBER 1993-II

Polaron

efFect on

the

binding energy

of

a

hydrogenic

impurity

in

a

quantum

well

C.

S.

Han and

T.

M.

Perng

Department ofElectrophysics, National Chiao Tung University, Hsin C-hu, Taiwan, Republic

of

China (Received 26 November 1991;revised manuscript received 14June 1993}

We have studied the polaron effect on the binding energy ofa hydrogenic impurity in aquantum-well

structure. The interactions ofan electron with both the confined bulk phonon and the interface phonon are taken into account. The competition between these two phonon modes isdiscussed. We have also extended the calculation for the case ofafinite quantum well. Itisfound that the polaronic correction becomes bigger as the potential barrier gets higher.

I.

INTRODUCTION

In recent years many theoretical and experimental in-vestigations have been performed on the issue

of

the "hy-drogenic" binding

of

an electron to a donor impurity in a semiconductor quantum well or heterostructure. Bas-tard' proposed a variational approach to calculate the binding energies

of

donor levels in terms

of

the well thickness and impurity positions in the quantum well as-suming the potential barrier at the interface to be infinitely high. Mailhiot, Chang, and McGill, Greene and Bajaj, and Liu and Quinn extended the work to cal-culate the binding energies

of

the ground state and several excited states

of

hydrogenic donors in

GaAs-Ga& Al As quantum well with finite barrier height. Later, Chaudhuri and Lane and Greene studied the hy-drogenic impurity states in multiple-quantum-well struc-tures. The effects

of

a finite-width barrier upon the bind-ing energies are discussed. Recently, the anisotropic effect and the quantum-confined Stark effect ' on the hydrogenic impurities in the quantum well have also been reported.

Since the

III-V

materials used in producing typical quantum-well structures are polar in nature, an electron weakly bound to a hydrogenic impurity in this system will interact with the longitudinal-optical phonons

of

the host semiconductor and tend toincrease the donor bind-ing energy. In the case

of

a bulk semiconductor, this po-laron effect is reasonably well understood on the basis

of

the Frohlich model.

"

The situation in a confined geometry, such as a quantum well, is considerably less clear. In the past several years, a number

of

works have been carried out to investigate the polaron effect in a quantum well system. Ercelebi and Tomak' studied the effect

of

electron-phonon coupling on the binding ener-gies

of

a hydrogenic impurity in a GaAs-Ga,

„Al

As quantum well and found that the correction becomes siz-able as the electron gets more deeply bound. Later, De-gani and Hipolito' calculated the polaron effect on the ground state as a function

of

electron density. The screening effect

of

the impurity potential is discussed. Mason and Das Sarma' have calculated the phonon-induced shift in the impurity binding energy due to electron-phonon interaction in a two-dimensional

quantum-well system.

It

is found that the polaron shifts in donor energy levels are

of

the order

of

10%%uo in the

GaAs-Ga&

Al„As

system. However, in all the calcula-tions mentioned above, the electron-phonon interaction is taken tobe

of

Frohlich type asin the bulk case. The pho-non confinement effect and the interface phonon are ig-nored. This is an oversimplification; indeed, several ex-periments' carried out recently indicate that the opti-cal phonon in the GaAs-Ga& Al As quantum-well sys-tem is confined,

i.e.

, the z-polarized (z perpendicular to the surface) optical phonons are equivalent to those vi-brations in the infinite crystal whose wave vector is given by mm/L, where

L

is the thickness

of

the well and m is an integer. The series

of

phonons labeled by m are termed confined phonons and also sometimes called

"folded"

phonons.

It

has been pointed out in various investiga-tions that phonon confinement effects lead to important modifications in the transport properties. Therefore in studying the polaron effect in a quantum-well structure, the confined bulk phonon should be taken into account. There is, however, another type

of

phonon which isalso quite important in the quantum-well system. This is the interface mode. Recently, the interaction

of

an electron with the interface phonon mode has been theoretically es-tablished for two-dimensional semiconductor heterojunc-tions.

'

Degani and Farias studied the exciton prob-lem in an AlAs/GaAs system and found that the inter-face phonon has significant effect on the exciton binding energy. Experimental observations

of

these interface modes are reported by Sood et

al.

' in backscattering Raman spectra, by Lambin et a/. in high-resolution electron-energy-loss spectra, by Meynadier et

al.

in

Ra-man scattering

of

high order at the resonance with the lowest optical transition, and by Gammon, Merlin, and Markoc in magnetic-field-enhanced Raman scattering. The purpose

of

this paper is to investigate the polaron effect on the hydrogenic impurity states in a quantum well by including both the interactions

of

the electron with the confined bulk phonon and the interface phonon. The competition between these two phonon modes and the correction on the binding energy

of

the hydrogenic impurity in the GaAs-Ga& Al As quantum well will be studied.

We shall also extend the calculation to the case

of

the

1993

finite quantum well.

It

has been pointed out recently'

'

that as the binding energy increases the localization

of

an electron becomes more pronounced, and this in turn in-creases the importance

of

electron-phonon coupling. Thus the polaron effect is expected to be smaller for the finite potential barrier where the binding would be

weak-er.

However, this important behavior has been only briefly speculated upon in previous works, and no calcu-lation was given since they considered only the case

of

the infinite quantum well.

II.

THEORY

H,

=

2m

2+

2

i/2+

( )2]1/2

(2) where _p

=x

+y

is the distance in the layer plane mea-sured from the impurity site, and z, is the coordinate

of

the impurity site along the superlattice axis. We first as-sume, for the sake

of

simplicity, that the barrier potential

V(z) has infinite height.

ized

)L/2

0,

Izl&L/2.

Later, we shall extend the calculations forthe finite quan-tum well. HIQ and

H,

Io are the Hamiltonians for the free interface phonon and its interaction with the elec-tron, respectively: Hgo

—

gftco s s q (4) —ql~ L/21+ —ql~+L/—_2I)( —qp +

_+H

e—IO

~

q q ~ ~ q (5) where s

q (sq)denotes the creation (annihilation) operator

for the IOphonon with the two-dimensional wave vector

q. For

a heterostructure, the frequencies

of

the interface modes, co+are determined by e,(co)

=

—

e2(co),with GaAs as medium 1 and Gai

„Al

As as medium

2.

The frequency-dependent dielectric function. isgiven as

s

e(co)

=e„+

1

—

e)'/AT

Let us consider a hydrogenic donor impurity atom lo-cated in a quantum well

of

thickness

L.

Within the framework

of

the effective-mass approximation, the Hamiltonian

of

this system contains five parts: electron-ic; free interface phonon (IO); electron —IO-phonon in-teraction; free bulk optical phonon (BO); and electron —BO-phonon interaction.

He

+

HIO

+

He—IO

+

HBO

+

He—BO

The electronic Hamiltonian operator

H,

can be expressed as

2

where

e,

and e are the static and high-frequency dielec-tric constants, respectively. cuT is the transverse

optical-phonon frequency. The electron —IO-phonon interaction strength

I

is defined as '

I

=

[2~a,

(A/2mco, )'

]'

qA (7a)

where

3

is the area

of

the surface and the electron —IO-phonon coupling constant

1/2

e m 1

2vre,

2'

co,

p,

(co,

)+p2

(co,)

(7b) with p(co)

=

1 2 COg CO CO_T 2 2 2 CO COL

—

_COT (7c) The Hamiltonians HBO and He BQ for the free BO

phonon and the electron —BO-phonon interaction can be written as

AcoL

W„=

—

[4rca (fi/2m co )'

]'

(10)

Here Vis the volume and the electron —_BO-phonon cou-pling constant

ab=e

[(I/e

)

—

(1/e,

)](m/2A coL )'/ .

We shall use the variational method proposed by Lee, Low, and Pines to treat our problem. In order to calcu-late the ground-state energy, we choose a trial wave func-tion as

ie&=i@,

&[N,

N,

&,

where ~

P, )

is the electronic part

of

the wave function for

the impurity in the infinite quantum well, which is taken the form as used in Bastard model

HBQ

Q~~Lbk

bk

k

—ik _p

H

BQ

g

Wk sin(k,z)(e bk

+H.

c.

) k

where AcoL isthe BO-phonon energy. bk and bk are, re-spectively, the creation and annihilation operators for the

BO

phonon with wave vector

k=(k,

k,

).

For

the confined bulk phonon, only limited values for the zwave vector are allowed, given by

k, =nor/L,

where n is an in-teger, since these phonons are confined within the quan-tum well. Recently Ren, Chu, and Chang studied the anisotropy

of

the optical phonon and the interface mode in a superlattice, and found that the interface mode and the confined mode with zero node (n

=1)

do not coexist in a quantum well for afixed value

of

in-plane wave vec-tor q. Therefore, the electron —BO-phonon interaction for the n

=1

mode should be excluded to avoid doubled counting. The interaction strength W& is given as

~y)=.

N

cos(k,

'

z)

exp

—

_7,

—

_[p

+(z

—

z;)

]'/,

~z~

&L/2

48 POLARON EFFECTON THEBINDING ENERGY OF

A.

. .

11967

Ui

=exp

i

P

—gs

s fiq

g—

bi+,_bi,A'k _p (13)

N isthe normalization constant, A,is avariational

param-eter, and

_k,

=ir/L.

N&Ni, & is the phonon wave

func-tion, with Nq and Nk the number

of IO

and

BO

phonons, respectively. In the low-lying state, N&Ni, &can be taken

as IO&oi,&,the phonon vacuum state.

First, we make the unitary transformation

Taking the expectation value

of H'

with the trial wave function given in

Eq. (11), i.

e.,

E=&y,

;O,

O„IH

Iy,

;o,

o„&

.

The variational parameters

F

and Gk are determined by the variational conditions

5E/5Fq

=5E/5Gk

=0,

which yield

where

P

isthe two-dimensional momentum defined as

g2q2 AP

.

_q

F,

=

—

r, r~+

~

+

'

(g+g,

')-

(19a)

P

=p

+giriqs+s

+giiik

bi+, _bi,

.

q k

The Hamiltonian istransformed into H1

=

U1+HU1

p2

+

V(z)+

P

—

gAqs+s

—

girik b„+bi,

2m 2m

L

+a~,

_ys,

+s,

_+yr,

_(s,

++s,

)

q q

+ficol

_/bi,

bi,

+/8'J,

(bi,

+bi,

) .

k k

(14)

(15)

Ak

AP k

Here

_g,

and _gb are defined by QiiiqF

=i),

P

q

QAk Gi,

_=ribs

k

r~=rq&&,

lexp( qlz

L/2I)

+

exp(

—

qlz

+I

/2I )I&,&

(19b)

(20a) (20b)

(21a)

The Hamiltonian finally becomes

H'=

_{U2+H1U2 . (17)}

Interms

of

the second Lee-Low-Pines transformation,

U,

=exp

g(F,

s, +F,

's,

)+g(G„b„++G„*b„)

.

(16)

re,

=

_IV/,&

y,

Isin(

k,

z)I

y,

&

.

(21b)

With the trial wave function _IP,& given in

Eq.

(12),the

closed forms

_{of rq}

and 8'k can be obtained. Substituting Eqs. (19a)and (19b)into

Eq.

(18),the ground-state energy

E

isobtained as 2 _N2g2

E=

+

2m 2E'

cos(2k,

z,)

1+

1+k1A,

k1k. 2z

1+k

A, cosh exp g2 g2 2 _g2

+

",

Il

—

_(„,

+„,

) _j

—

yr',

r,

+'&

+

"&(„,

+„)

2m'

2m mX

Ak

A'k

—

_g

_{Wq ficoi}

₊

₊

_{(r), +r)b}

')

k Zm mk

(22)

For

slow electrons, Eqs. (19)and (20)yield 2x

9s

_1+2x,

+2xb

(23a)

E~

=Eo

—

E,

(25)

The binding energy

of

the hydrogenic impurity is given by where 2xb

1+2x,

+2xb

AP

.

_q

x,

=

_P

Qr

_m 2 g2 2

+

2m —3 (23b) (24a)

where Eo is the ground-state energy for an infinite quan-tum well without impurity. Minimizing

E

with respect to

A,_,we obtain the binding energy as a function

of

the well

width

L

and impurity position

z;.

Next, we shall consider the case

of

finite quantum well. The potential well isgiven as

2 m AP

k

xb

=,

XIV''

p

m /2k 2 P flML

+

2m (24b)

Vo,

Iz &

L

/2

.

(26) '

III.

RESULTSAND DISCUSSION A. Infinite quantum well

We first give the resu1ts o

f

the infinite quantum well for efine Al As system.

It

isconvenient to de ne

/

dth

ff

t

ber

R

*

=e

/2eoa

'.

The material parameters use in the calculation are given in a e

of

the binding energies as a function o ay-the variations o

t

e in '

d on-ed e impurities. er thickness for the on-center anu on-e ge im

TABLE

I.

The matenal paramrameters for GaAs and

Gal—xAlxAsused inthe calculations.

GaAs Ga& Al As Ep 12.9 10.9 0.14 12.04 10.57 0.30 11.18 10.16 0.36 10.89 10.04 (cm ') 293 285 278 275 coT (cm ~) 269 267 265 264 a Al As heterostructure is distributeded about GaAs-Ga&

„A

s e e

60/

the conduction

40%%uo on the valence band and o on

ce AE between band with thee total band-gap difference e w

the Al con-GaAs and Ga_a) Al As given as a function

of

t

It

is well known that when the potential well is ni e,

e - not be solved for assimply as

the bound-state energies canno

ne has tosolve ecase

of

the infinite quantum well. One has toso ve

.

W h 11use a general per-tal e uations. es a

b

hodp

rooposed by Lee an ei o

fini'te potential-well problems.

It

is foun a hen one is intereste h d is sufficiently accurate w-en

met o is su

Recently, this

ap-1 in the low-lying bound states.

as _pp d

t

d the anisotropic effect

as been appliea to s u y

cessful results have been obtained. ' _In

_t

_{is me o}

-state ener ies o

f

thefinite quantum well can be ap-the ei enenergies o

f

the infinite quantume proximated by 'g

'

n which de ends on the

11with a perturbative correction w ic epe inverse square root

of t

e po en i

11which must wave function o

f

the infinite' quantum we w ic

t e hard wall, the actual wave function wilill

11 th

enetrate a distancee 5 into the so t

wa,

ction

of

e size

of

the wave function by a frac ion

o ld

ob'o

1 b

26/L.

However, this increase cou

1 shifting the infinite wall farther away

y

py

~

6A

h m the original position by a distance

.

s e from eor'g'

d with the eigenenergies

of

the tial height islarge compare wi

low-lying sta es,

t

the penetration depth 6is given by

(0) 1/2

2m ( Vo

E)—

g2

where

E'

' represents the ground-state energy

of

the

cor-h a well width

L

can be considered the same as well with a we wi

well with a broadened well that in an infinite quantum we wi a

width

L

+26.

{a)

rity 3 UJ I I I I I I 4 1 0 I 2 3 L/a

"

I i I 8 9 10

(b)

3 LLj ge impurity 2 I i I i I I i I i I 0 I I I I i I I 5 6 7 8 9 0 1 2 3 4

L/a"

10

E/R*

as a function ofthe FIG~ 1.~ Reduced binding energy

E/

uced well width

L/a

for (a)the on-center impurit' yand (b) e - ' ' curve 1 (2) corresponds to the case the on-edge impurity. The curve cor

with (without) electron-phonon interactions.

'

h ut the electron-phonon interactions are n in the figure for comparison.

It

can e seen also shown in e

h th olaron effects on the binding energ

t

att

epoa

For

the on-center impurity the effecects ares about portant. or e

reduce to about

1.

8%

22%%u for the small well widths, and reduce

as the well width increases to

L/a

=20,

andand eventually a roaches the value

of

the bulk.

For

the on-edge

impur-it,

the polaron effect is arger an in the bulk limit to a out o

in ' b 23%%uo for very narrow well size.

ola-1bi and Tomak' have calculated the po

a-ron effec

t

foror the on-center impurity, and o aine in the bulk case and

15%

for the smamall well size. 3/o

int

e u

A ehave mentioned before, in these previous cal-ness. As we ave m

th honon confinement effects is ig culations e _p

e is also neglected. This is

of

the interface phonon mode is also neg ec e ~

course an oversimplification.

.

In our calculations, bot t e k n phonon confinementon and the 'interface _Phonon are ta en obe more re i-1

0

t.

Therefore, our result seems to into accoun

.

able than the others. We have also studied the corn eti-_p bulk honon and the interface tion between the confined _p

d

F'

re 2 gives the variation o in ing P

f

'

n

of

well width, including ony energy as a unctio

POLARON EFFECTON THEBINDING ENERGY OF

A.

. .

11969

(a)

ter impurity

Gal

„Al

As have been obtained with a different Al con-centration

x

from the Kramers-Kronig dispersion analysis

of

the infrared reAectivity spectra. Table

I

lists the values that are used in our calculations. We have used the general perturbative method

of

Lee and Mei to study the finite quantum-well problem. In order to check the validity

of

this method for our problem, apreliminary calculation is first performed for the case without the

I ~ l & I I I i I l l i l ) I ( I 0 1 2 3 4 5 6 7 8 9 10 L/a X=0.36 on-edge impurity 0 I I t 1 I I 0 1 2 3 4 5 6 7 8 9 10

L/a

0 a l s l t I ) l l I I I ~ I t l I I 0 1 2 3 4 5 6 7 8 9 10

L/a"

3-

X=0.30

FIG.

2. Reduced binding energy

E/R

*as a function ofthe reduced well width

L/a*

with the confined bulk phonon (curve

1)orthe interface phonon (curve 2)for an impurity located (a)

at the center or(b)on the edge ofthe well.

Ct

2-LU

found that the competition between these two modes in-dicates that in the case

of

small well thickness the inter-face phonon plays the dominant role and the confined bulk phonon makes little contribution. As the well width increases, the interface phonon contribution decreases and eventually the bulk phonon becomes the important one as

L

)

10a*.

This is in accordance with our expecta-tion. Since the bulk phonon is confined in the longitudi-nal direction (i.

e.

,the z axis) there should be no bulk pho-non effect as the well width

I,

approaches zero.

It

is worth to note that this result is different from the previ-ous calculations where the usual Frohlich interaction was used and the phonon confinement effect was neglected so that the main contribution still came from the bulk pho-non even as

I

—

+0. In our calculations, it isthe interface phonon which makes the main contribution as the well width gets smaller. Tatham et

al.

have recently report-ed a significant increase in the relaxation rate for a very narrow well (25 A), which we believe is responsible for the interface modes.

B.

Finite quantum well

In this section, we present the results for the case

of

the The

TO

LO for

p 0 1 I 1 I I I 2 3 4 5 6 7 8 9 10 L/a~ X =0.14 CL

2-LU 0 I t I I I I I I l 0 1 2 3 4 5 6 7 8 9 1Q L/a~

FIG.

3. The variation ofthe reduced binding energy

E/R*

without the electron-phonon interactions as a function ofthe

well width

L/a*

using the general perturbative method (dotted

line) and the exact method of Liu and Quinn (solid line) for several values of the barrier heights (a) Vo(x

=

0.36), (b)

Vo(x

=0.

30), and (c) Vo(x

=0.

14). The impurity is at the of

2.7 2. 6-2. 5-2.4 2. 3-2. 2-2. 1-2 CL lg-LU1. 8-1.7— 1, 6-1. 5-1. 4-1. 3- 12-1, 1-I I I I I 2 4 6 L/a" (a) I I 8 10 3,1 3.2— 2. 8-1. 8-1.2— 1 0 I i 1 I I I I I 2 4 6 8 10 L/a" 2.8 2. 6-2. 4-2. 2-1. 8- ~1.6-LU1. 4-1. 2-0. 8-0. 6- 04-02' (a) I I I I 2 4 6 8 10 L/a" 3.4, 3. 2-2.8— 2.6 2g-2.2 2-lZ 1. 8-LU1.6 1. 4-1. 2-1. 1-0.8 0.6 0.4 0.2 I I I I I I I 2 4 6 8 10 L/a" 3. 4-3. 2-2.8 CL 2.2 LU 2 1.8 1.2 1 0 2 4 6 L/a" (c) 10 3.6 3. 4-3. 2- 3-2.8 2.6 2.4 2.2 CL LU 1. 8-1. 6-1. 4-1. 2-0.8 0.6 0. 4-0.2 (c) I I I I 2 4 6 8 10 L/a"

FIG.

4. Reduced binding energy

E/R

*as a function ofthe re uced well width

L/a

for an impurity located at the center

in the fin'ite quantum well. Curve 1 (2)correspon s o

with (without) electron-phonon interactions or Vo(x

=0.

14),(b) Vo(x

=0.

30),and (c)Vo(x

=0.

36).

FIG.

5. Same asin Fig. 5but forthe impurity located on the

edge ofthe well.

0.5(

Q.4—

electron-phonon interaction, andd the results are com-d to those

of

the exact calculation using the method

pare o o

of

Liu and Quinn as shown in Fig.

3.

It is clear y n as the well width

t

ha

t

the agreemente is quite good even as t

d s to

L

/a *

-0.

3 for the case

of x

=

=0.

36 reduces to a

r

x

=0.

14.

L/a*-0.

4 for

x

=0.

30, and

L/a*-0.

65 for

x

=

Therefore, our method is reasonably applicable to the problem when the well size is greater than the above re-gions. This method is then used to study the case with the electron-phonon interactions taken into account.

We have calculated the binding energy as a function o well thickness for different potential barrier heights

d

t

=0.

14

0.

30, and

0.

36.

The results corresponding to

x

=

~

F

0

are shown in

Fi

.

4forthe on-center impurity and in ig. 5 for the on-edge impurity. Figure 6gives the correction tions. Our results show that the polaronic shift is a so quite impor ant

t

for the case

of

the finite quantum we

.

For

the GaAs-Ga07Alo 3As system

(x

=0.

3),

which cor-res onds to the potential-well height

V0=36R

*,

the po-respon s o

laronic shift ranges from about 13%%uo forsmal1weell size to

8/ f

the bulk limit.

It

is interesting to note from

Fi

. 6 that the polaron effect becomes bigger as e

p-tentialig. barrier gets higher. This isbecause

t

'g

a

~ ~ ~

he hi her the barrier' the larger the bindinge energy, the localization

of

the electron becomes more pronounce~ an us creases the importancee '

of

the electron-phonon interac-tion. As we have pointed out before, this impor an behavior has only been briefly speculated upon in

previ-LLI 0,2— :x =0.36 :x =0.30 :x=0.14 0.1 1 s I r I i I i I s 1 l I i I s I 6 7 8 9 10 0 1 2 0.5, Q4— 0.3— Ii) &j 0.2— 0.36 0.30 0.14 0 1— 0.0 0 I I i I I I ss I i l i I s l 1 3 4 5 6 7 8 9 0

L/a"

FIG.

6. The shifts ofbinding energy due to the electron-phonon interactions for (a)on-center and _{b) on-ed e impurities}- _g in the finite quantum well, with different Vo corresponding to x

=0.

36(curve 1),0.30(curve 2), and 0.14 (curve 3).

POLARON EFFECTON THEBINDING ENERGY OF

A.

. .

11971

ous works, '

'

but no calculation was given since they only considered the case

of

the infinite quantum well. Our work is an explicit calculation and presents the re-sults for different potential barrier heights.

For

example, for the on-center impurity with

I. =a*,

the percentage

of

polaronic correction is

10%

for the barrier height Vo(x

=0.

36),

and decreases to about

7%

for Vo(x

=0.

14).

It

is clear to see that, in general, for a lower quantum-well height where the binding energy is smaller, the polaron effect becomes weaker.

It

is also worth noting that the tendency

of

decreasing polaron effect as the barrier is lowered is quite fast for smaller well thicknesses. As the well width becomes larger, the polaronic correction is almost the same for different bar-rier heights, and approaches the limiting bulk value.

IV. CONCLUSION

We have studied the polaron effect on the hydrogenic impurity in the GaAs-Ga& Al As quantum-well sys-tem.

It

is found that the shifts

of

binding energy due to electron-phonon couplings are quite important for both the on-center and on-edge impurities. The polaronic correction can be as large as 23%%uo for the small well

thickness. In this work both the interactions

of

the elec-tron with the confined bulk phonon and interface phonon are taken into account. Therefore, our result seems tobe more reliable than those

of

previous calculations where the phonon confinement effect and the interface phonon mode are neglected. The competition between the inter-face and confined bulk phonons is also investigated. Our results show that the dominant contribution comes from the interface mode in the case

of

thin layers, and the bulk phonon is more important as the well thickness becomes larger than

10a*.

We have extended the work for the case

of

the finite quantum well with different potential barrier heights.

It

is found that the higher the quantum-well barrier where the binding energy is larger, the larger the polaronic correction due tothe increasing importance

of

the electron-phonon interaction. We also find that the polaron effect decreases very quickly as the barrier height is lowered, for asmall well width and approaches the lim-iting bulk result forthe large well size.

ACKNOWLEDGMENT

The work was supported by the National Science Council

of

Taiwan.

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